November 2022 Local scaling limits of Lévy driven fractional random fields
Vytaut˙e Pilipauskait˙e, Donatas Surgailis
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Bernoulli 28(4): 2833-2861 (November 2022). DOI: 10.3150/21-BEJ1439


We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields X on R2 written as stochastic integral with respect to infinitely divisible random measure. The scaling procedure involves increments of X over points the distance between which in the horizontal and vertical directions shrinks as O(λ) and O(λγ) respectively as λ0, for some γ>0. We consider two types of increments of X: usual increment and rectangular increment, leading to the respective concepts of γ-tangent and γ-rectangent random fields. We prove that for above X both types of local scaling limits exist for any γ>0 and undergo a transition, being independent of γ>γ0 and γ<γ0, for some γ0>0; moreover, the ‘unbalanced’ scaling limits (γγ0) are (H1,H2)-multi self-similar with one of Hi, i=1,2, equal to 0 or 1. The paper extends Pilipauskait˙e and Surgailis (Stochastic Process. Appl. 127 (2017) 2751–2779) and Surgailis (Stochastic Process. Appl. 130 (2020) 7518–7546) on large-scale anisotropic scaling of random fields on Z2 and Benassi et al. (Bernoulli 10 (2004) 357–373) on 1-tangent limits of isotropic fractional Lévy random fields.

Funding Statement

VP acknowledges financial support from the project ‘Ambit fields: probabilistic properties and statistical inference’ funded by Villum Fonden.


The authors thank two anonymous referees and the AE for useful comments.


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Vytaut˙e Pilipauskait˙e. Donatas Surgailis. "Local scaling limits of Lévy driven fractional random fields." Bernoulli 28 (4) 2833 - 2861, November 2022.


Received: 1 February 2021; Published: November 2022
First available in Project Euclid: 17 August 2022

MathSciNet: MR4474564
Digital Object Identifier: 10.3150/21-BEJ1439

Keywords: Fractional random field , Lévy random measure , local anisotropic scaling limit , multi self-similar random field , rectangular increment , scaling transition


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Vol.28 • No. 4 • November 2022
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